# Find the smallest eigenvalue (not absolute value ) for a generalized eigenvalue problem

I tried to find the smallest eigenvalue for a generalized eigenvalue problem A c= \lambda B c (the default option is by absolute value) by

a = {{1, 2, 0}, {2, 5, 3}, {0, 3, 9}}; b = {{1, 5, 3}, {5, 2, 0}, {3, 5, 0}};

n = 5; dimension = 3; Eigenvalues[N[{a, b}], 3]

-n + Eigenvalues[ N[{ a + n IdentityMatrix[dimension], b + n IdentityMatrix[dimension] }, 32 ] ]

It does not work. How to find smallest eigenvalue for a generalized eigenvalue problem?

P.S. In my case (before shifting N* Identity matrix), all eigenvalues are real.

• One approach is to get an a priori estimate of largest in magnitude, and shift along positive real axis with a multiple of identity matrix so that the most negative eigenvalue becomes also the largest. This will only work if there are no complex ones that become larger still. Jan 22, 2015 at 16:13
• I tried to shift by n IdentityMatrix[dimension] in both matrices $a$ and $b$, seems this prescription only works in eigenvalue not generalized eigenvalue problem.. Jan 22, 2015 at 17:35
• If your eigenvalues are complex, how would you define the "smallest"--magnitude? Jan 22, 2015 at 18:07
• I guess I had not realized the general case would behave differently. I sent a link to the question to someone more knowledgeable, so maybe a better answer will emerge. Jan 22, 2015 at 19:02
• Does this help? Jan 23, 2015 at 4:51